Optimization in container liner shipping

Judith Mulder1 and Rommert Dekker1

1Econometric Institute, Erasmus University Rotterdam, the Netherlands

Econometric Institute Report 2016-05

Abstract

We will gives an overview of several decision problem encountered in liner shipping.

We will cover problems on the strategic, tactical and operational planning levels as well

as problems that can be considered at two planning levels simultaneously. Furthermore,

we will shortly discuss some related problems in terminals, geographical bottlenecks for

container ships and provide an overview of operations research methods used in liner shipping

problems. Thereafter, the decision problems will be illustrated using a case study for six

Indonesian ports.

1 Introduction

Seaborne shipping is the most important mode of transport in international trade. In comparison

to other modes of freight transport, like truck, aircraft, train and pipeline, ships are preferred for

moving large amounts of cargo over long distances, because shipping is more cost e�cient and

environmentally friendly (Rodrigue et al. 2013). Reviews of maritime transport provided by the

United Nations Conference on Trade And Development (UNCTAD 2014) show that about 80%

of international trade is transported (at least partly) by sea. Sea transport can be separated

into dry bulk (e.g. steel, coal and grain), liquid bulk (e.g. oil and gas) and containerized cargo.

In 2013, containerized cargo is with a total of 1.5 billion tons responsible for over 15% of all

seaborne trade, which resulted in a world container port throughput of more than 650 million

twenty-foot equivalent units (TEUs).

The shipping market comprises three types of operations: tramp shipping, industrial shipping

and liner shipping (Lawrence 1972). Tramp ships have no �xed route, but ensure an immediate

delivery for any type of cargo from any port to any port, resulting in irregular activities. The

behaviour of tramp ships is thus comparable to taxi services. In industrial shipping the cargo

owner also controls the ships used to transport the freight. The objective of industrial operators

is to minimize the cost of shipping the owned cargoes. Liner ships follow a �xed route within a

�xed time schedule and serve many smaller customers. The schedules are usually published online

and demand depends on the operated schedules. Hence, liner shipping services are comparable

to train and bus services.

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In the next section, we will discuss a variety of liner shipping problems on the strategic, tactical

and operational planning levels, while Section 3 introduces some problems related to terminal

operations. Sections 4 and 5 discuss respectively the in�uence of geographical bottlenecks and

the importance of operations research in solving these problems. This overview is based on the

following overview articles: Ronen (1983, 1993), Christiansen et al. (2004, 2007, 2013), Meng

et al. (2014). In Section 6, a case study is performed for six Indonesian ports to provide insight

into the di�erent problems. Finally, conclusions are drawn in Section 7.

2 Container liner shipping

We will focus on the liner shipping operations concerned with the transport of containers. Liner

shipping operators face a wide variety of decision problems in operating a liner shipping network.

First, at the strategic planning level, the �eet size and mix problem and the market and trade

selection problem need to be solved. In the �eet size and mix problem, operators decide on the

�eet composition and in the market and trade selection problem on which trade route to serve.

At the tactical planning level the network needs to be designed, prices need to be set and empty

containers have to be repositioned. Finally, at the operational level, operators need to determine

the cargo routing through the network and how to deal with disruptions. Furthermore, they can

make adjustments to the earlier set prices and need to determine a plan to store all the containers

on the ship during the loading process. These problems are considered in respectively the cargo

routing, disruption management, revenue management and stowage planning problems. Some

problems have to be considered at both the tactical and the operational planning level, such as

setting the sailing speed and optimizing the bunkering decision and designing a (robust) schedule.

In this section, we will introduce all these decision problems. In these problems we will make use

of the following terminology. Liner shipping operators will also be referred to as liner shipping

companies, liner companies or liners. Liner ships follow �xed routes, which are sequences of port

calls to be made by the ship. Route networks consist of a set of services, which are routes to

which a ship is allocated. Besides publishing their route networks, liner companies also publish

the exact arrival and departure days at each port of call. When we refer to the route together

with the arrival and departure days, we will talk about a schedule. Finally, a round tour refers

to one traversal of a route and a (sea) leg refers to the sailing between two consecutive ports.

2.1 Strategic planning level

The strategic planning level consists of long term decision problems. Generally, these problems

are only solved at most once a year. Examples of long term decision problems in container liner

shipping are: the �eet size and mix problem and the market and trade selection problem.

2.1.1 Fleet size and mix

In the �eet size and mix problem, a liner company decides on how many ships of each type to keep

in its �eet. Container ship sizes have increased substantially because of the growth in container

trade and because of competitive reasons. For example, the Emma Maersk (introduced in 2006)

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has an estimated capacity of more than 14,500 TEU. Before the introduction of the Emma

Maersk, the capacity of the largest container ship in the world was less than 10,000 TEU. In

2013, Maersk introduced a series of ships belonging to the Triple E class with capacities of over

18,000 TEU, while both MSC and CSCL introduced container ships with a capacity of more than

19,000 TEU in 2015. Ships bene�t from economies of scale when they are sailing at sea, but they

might su�er diseconomies of scale when berthing in ports. However, the e�ect of the economies

of scale at sea is much larger than the e�ect of the diseconomies of scale in ports (Cullinane

and Khanna 1999). Hence, economies of scale in larger container ships can lead to substantial

savings if the capacity of the ship is adequately used. However, if the demand decreases and the

liner company is not able to �ll these large ships any more, higher operational costs are incurred

by these large ships. Therefore, �eet size and mix problems are used to balance the possible

bene�ts from economies of scale with the risk of not being able to use the full capacity of the

ships. Since building a new container ship may take about one year and ships usually have life

expectancies of 25-30 years, future demand and availability of ships play an important role in the

�eet size and mix problem. Pantuso et al. (2014) present an overview of research conducted on

the �eet size and mix problem. Most of these works incorporate ship routing and/or deployment

decisions in order to ensure feasibility of demand satisfaction and capacity constraints.

2.1.2 Market and trade selection

Before a liner container shipping company starts building a network and operating the routes,

it has to decide which trade lanes to participate in. The Asia-Europe trade lane is an example

of a popular trade lane. Clearly, the selected trade lanes in�uence the type and number of ships

required. For example, trade lanes serving the US will usually not use vessels from the Maersk

Triple E class, since they can not sail through the Panama Canal and most ports in the US are

not capable of handling these large vessels. Furthermore, the type and amount of cargoes that

have to be transported and the required sailing frequency may in�uence the ship types used on

the trade lane.

2.2 Tactical planning level

Medium-term decision problems belong to the tactical planning level. Liner companies usually

change their service networks every 6-12 months, but more often in case of worldwide disruptions.

Problems that have to be solved again each time the service network is adjusted are considered to

belong to the tactical planning level. The examples that will be discussed next are: the network

design problem, the pricing problem and the empty container repositioning problem.

2.2.1 Network design

The network design problem in liner shipping can be split into two subproblems. The �rst

subproblem is the routing and scheduling problem, which is concerned with determining which

ports will be visited on each route, in which order the ports will be called at and what the arrival

and departure times at each port will be. Many studies only consider the routing decisions in the

network design problem and do not address the scheduling problem of determining the actual

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arrival and departure time. The second subproblem considers the �eet deployment and frequency.

Here, the liner company determines which ships will be used to sail each route and with what

frequency the ships will call at the ports along the route. In general, a weekly frequency is

imposed, which facilitates planning by shippers, but this can be relaxed to a biweekly frequency

for low demand routes or multiple port calls per week for high demand routes. Sometimes sailing

speed optimization is considered as a third subproblem of the network design problem, but in

most studies the sailing speed is either assumed to be �xed and known, or will follow directly from

the imposed frequency. Usually, the cargo routing problem is already included (using expected

demand as input) in the network design problem in order to evaluate the pro�tability of a

service network. The cargo routing problem will be discussed in more detail with the operational

planning level problems.

The structure of the routes in a network can be divided into several types, like non-stop services

or end-to-end connections, hub and spoke systems, hub and feeder systems, circular routes,

butter�y routes, pendulum routes and nonsimple routes (?Brouer et al. 2014a). A non-stop

service or end-to-end connection provides a direct connection between two ports: a ship sails

from one port to the other and immediately back to the �rst port; sometimes this is called a

shuttle service, although that also requires a high frequency. In a hub and spoke system, usually

one port is identi�ed as the main or hub port. All other ports (also called feeder ports) are

served using direct services from the hub port. However, it is also possible that multiple hubs

are applied, which are connected with each other and used as transshipment ports to satisfy

demand between di�erent feeder ports, in which case they might also be referred to as main

ports. In the hub and feeder system, feeder ports might also be visited on routes with multiple

port calls. Circular routes are cyclic and visit each port exactly once, while butter�y routes

allow for multiple stops at the same port in one cycle. Pendulum routes visit the same port in

both directions, only in reverse order. Finally, ports can be visited multiple times on nonsimple

routes. Examples of some of these route types are provided in the case study in Section 6.

The liner shipping network design problem has attained quite some attention in the literature.

We will brie�y describe some of the recent publications on this problem. Plum et al. (2014b)

consider a subproblem of the network design problem. They develop a branch-and-cut-and-price

algorithm to �nd a single vessel round trip. Each port has to be visited exactly once and the

best paying demand pairs are accepted and transported. Polat et al. (2014) consider an adapted

neighbourhood search method to solve a hub and feeder system with one single hub. Finally,

Zheng et al. (2015) propose a genetic algorithm to solve the same problem with multiple hubs.

Wang and Meng (2014) propose a column-generation heuristic approach to �nd the best liner

shipping network. Each port can be visited twice during each route: once on the inbound

direction and once on the outbound direction. Brouer et al. (2014a) provide both a base mixed

integer programming formulation for the network design problem and benchmark data instances.

They propose a column generation approach to generate butter�y and pendulum routes. Plum

et al. (2014c) extend the butter�y routes as used in the benchmark model to routes with multiple

butter�y ports. Brouer et al. (2014b) propose a matheuristic to solve the base network design

problem with nonsimple routes. Although their assumptions are a bit more restrictive than in

the benchmark paper (Brouer et al. 2014a), they are able to construct a more pro�table route

network using this approach.

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Liu et al. (2014), Mulder and Dekker (2014) and Wang et al. (2015) consider slightly di�erent

network design problems. Liu et al. (2014) consider a problem in which the port-to-port demand

is combined with the inland transportation. They start with an initial liner network and try

to improve it while also including the transportation between the ports and the real origin

and destination of the demand. Mulder and Dekker (2014) consider the strategic liner shipping

network design problem, including the �eet size and mix problem, using a hub and feeder network

structure. Wang et al. (2015) consider the liner shipping network alteration problem. In this

problem, an initial liner network is given and this network is modi�ed to become more pro�table.

2.2.2 Pricing

The goal of liner companies is to maximize pro�t by transporting containers from one port to

another. The revenue of the company is determined by the amount of containers that are trans-

ported and the price that will be charged for each container. The pricing problem is concerned

with which price to charge for each possible demand pair. Factors that in�uence the price are

for example: distance, trade direction, expected demand and expected capacity. The pricing

problem is more a marketing, micro-economic problem than an operations research problem. Al-

though it is an interesting problem, it has hardly been touched. Two approaches exist: cost-plus

and what the market can pay. Yet, even determining the cost is a di�cult allocation problem.

2.2.3 Empty container repositioning

Containers delivering import products in a region can be re-used to transport export goods to

another region. However, most regions face an imbalance between import and export containers.

This trade imbalance results in an excess of empty containers in regions with more import than

export and a shortage of containers for high export regions. The empty container repositioning

problem tries to reallocate the empty containers in order to solve the imbalance, where costs are

associated with transporting a container from one region to another. The repositioning of empty

containers is considered to be very costly, since there is no clear revenue associated with it.

Some recent papers dealing with the empty container repositioning problem are the following.

Both Di Francesco et al. (2014) and Long et al. (2015) consider the empty container repositioning

problem under uncertain container demand and use a stochastic optimization approach to solve

it. Zhang and Facanha (2014) consider the problem of repositioning empty containers to the

location of demand in the US. Empty containers are transported using trucks or trains to the

location where they can be loaded. If containers can not be allocated to a loading location, they

are transported to a West Coast port to be shipped to Asia. Huang et al. (2015) consider the

network design problem with both laden and empty container repositioning. Multiple hub ports

are identi�ed, where transshipment from feeder ports might take place. They select the best

routes from a candidate set of routes, which is used as input in the model.

2.3 Operational planning level

The operational planning level captures the problems that occur during the execution of the

routes in the service network. In order to solve operational level problems, reliable information

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about the actual situation is needed. Hence, operational problems usually need to be solved

relatively shortly before the solutions have to be implemented. Next we will discuss the cargo

routing problem, the disruption management problem, the revenue management problem and

the stowage planning problem.

2.3.1 Cargo routing

The cargo routing problem takes the liner shipping network and container demand as an input.

The goal of this problem is to �nd a cargo �ow over the network, satisfying the capacity con-

straints imposed by the allocated container ships, that maximizes the pro�t of transporting the

containers. Costs are associated to (un)loading and transshipment operations. A transshipment

occurs when a container has to be unloaded from a ship and loaded to another ship in order to

arrive at its destination. Additionally, penalties can be imposed for demand that is not met.

It is also possible to include transit time constraints to guarantee that containers will arrive on

time at their destination.

Formulations of the cargo routing problem can be distinguished in OD-based link �ow formu-

lations, origin/destination-based link �ow formulations, segment-based �ow formulations and

path-based formulations (Meng et al. 2014). All �ow formulations consider the amount of �ow

at a link or segment of the route as decision variables in the model. Flow balance constraints

ensure that all �ow starts at the origin port and arrives at the destination port, but the exact

route followed by a container might not be immediately clear from the model. In the OD-based

link �ow formulation, both the origin and the destination of the container are stored for each link

in the network, while the origin/destination-based link �ow formulations only store the origin

or destination of the container. In this way, the number of decision variables can be reduced

signi�cantly. In a segment-based �ow formulation, consecutive links of a route are already com-

bined into segments before building the model. Segment-based �ow formulations reduce the

number of decision variables even more, but limit the possibility of transshipment operations to

the ports at the beginning and end of the prede�ned segments. Finally, in path-based formu-

lations complete container paths from origin to destination are generated beforehand and used

as decision variable in the model. These paths might also include transshipment operations.

The disadvantage of path-based formulations is that the number of paths might explode, such

that more complex methods, like column generation, are needed to solve the problem. However,

path-based formulations can usually be solved faster than �ow-based formulations. Furthermore,

transit time constraints are easily incorporated in path-based formulations, while this is gener-

ally much more troublesome in �ow-based formulations. Little research is performed on the

separate cargo routing problem: usually it is considered as a subproblem of the network design

problem. Recently, Karsten et al. (2015) considered the cargo routing problem with transit time

constraints. They propose a path-based formulation exploiting the ease to include transit time

in this type of model. Their �ndings show that including transit time constraints in the cargo

routing model is essential to �nd practically acceptable container paths and does not necessary

increase computational times.

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2.3.2 Disruption management

During the execution of the route schedules, ships may encounter delays. The disruption man-

agement problem focuses on which actions should be taken in order to get back on schedule

after a disruption has occurred. Examples of actions that might be performed are: changing the

sailing speed, swap port calls, cut and go (leave the port before all containers are (un)loaded)

or skip a port. Usually, the goal of disruption management is to �nd a sequence of actions with

minimum cost such that the ship will be back on schedule at a predetermined time. Brouer et al.

(2013) propose a mixed integer programming formulation to solve this problem and prove that

the problem is NP-hard. However, experimental results show that the model is able to solve

standard disruption scenarios within ten seconds to optimality.

2.3.3 Revenue management

At the operational level, more information about the demand and available capacity of a ship is

available. Therefore, it might be pro�table for liner companies to vary their prices based on the

available capacity between a port pair. Liners will probably charge higher prices related to low

capacity pairs, while they might reduce the prices on legs where the available capacity is high.

2.3.4 Stowage planning

The stowage planning problem determines at which location containers are stored on the ship

during the loading process. The stowage planning is a very complicated process with many

constraints. Essential constraints are for example the stability of the ship both during the next

sea leg and during the (un)loading process. Furthermore, containers may have to be stored at

speci�c locations on the ship, like reefer containers. However, the storage of the containers also

in�uence the (un)loading process in the next ports. Ideally, all containers with destination in

the next port are stored on top of the stack, but this may take too many movements in the

current port. Hence, a trade-o� between the number of moves required to store and to discharge

a container has to be made. Tierney et al. (2014) prove that the container stowage planning

problem is a NP-complete problem.

2.4 Both tactical and operational level

Finally, some problems can either be considered at two di�erent planning levels or have to be

considered at two planning levels at the same time. For example, sailing speed optimization

and bunkering optimization can be considered at the moment a new service network is designed,

but the solutions to these problems can be reconsidered during the execution of the routes.

Furthermore, robust schedule design is an example of a decision problem that combines decisions

to be taken at the tactical and at the operational level.

2.4.1 Sailing speed and bunkering optimization

Both sailing speed optimization and bunkering optimization are typical problems that can be

considered at two di�erent planning levels. At the tactical planning level, the environmental

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aspects of sailing speed are usually considered. At the operational level, sailing speed is mostly

used as an instrument to reduce delays incurred by the ship. The bunkering optimization problem

is concerned with deciding at which ports ships are going to be refuelled. Initially, a bunker

refuelling plan is made given estimates of the bunker price at the moment the ship will be at a

bunkering station. Shipping lines also regularly make bunkering contracts, containing the ports

where bunker can be purchased, the amount to be purchased, the price to be paid and the validity

duration of the contract (Pedrielli et al. 2015). However, due to �uctuations in prices or fuel

consumption, this initial plan might have to be adjusted at the operational planning level. At

this stage, more accurate information about the fuel prices and availability at the ports and the

bunker level of the ship is available. Sailing speed plays an important role in the bunker fuel

consumption of ships and hence sailing speed optimization is often included in the bunkering

optimization problem (Yao et al. 2012).

Recently, the sailing speed and bunkering optimization problems have received increasing atten-

tion. Psaraftis and Kontovas (2014) consider the speed optimization problem at the operational

planning level. They include fuel prices, freight rates, cargo inventory costs and fuel consump-

tion dependencies on payload into their model. Kontovas (2014) and Du et al. (2015) consider

the in�uence of sailing speed on fuel emissions. Wang et al. (2013) provide a literature review

on bunker consumption optimization problems. Bunker consumption is an important input for

bunkering optimization. Yao et al. (2012) study the bunker fuel management strategy for a

single liner shipping route. The strategy consists of the selection of the bunkering ports, the

determination of the bunkering amounts and the adjustments in the sailing speeds. They con-

sider a deterministic situation in which all parameters, including bunker costs, are �xed and

known. Plum et al. (2014a) and Pedrielli et al. (2015) study the problem to determine the op-

timal bunkering contracts. Plum et al. (2014a) propose a mixed integer programming model,

which is solved using column generation. Also, the possibility to purchase bunker on the spot

market is included in their model. Pedrielli et al. (2015) use a game theoretical approach to

design the contracts. Wang et al. (2014) propose a fuzzy approach to include uncertainties in

the bunkering port selection problem. Their method returns a ranking of ports based on the

pro�tability to bunker in these ports. Sheng et al. (2015) implement an (s, S) policy to jointly

optimize the bunkering and speed optimization problem taking into account both bunker price

and consumption uncertainty. Finally, Wang and Meng (2015) consider the robust bunker man-

agement problem, taking into account that the real sailing speed might di�er from the planned

sailing speed.

2.4.2 Robust schedule design

Robust schedule design can be seen as a combination of the scheduling problem at the tactical

level and disruption management or sailing speed optimization at the operational level. The

order in which ports are visited is considered to be an input of this problem. The goal is to

jointly determine the planned arrival and departure times in each port and the actions that will

be performed during the execution of the route when delays are incurred. The di�culty of this

problem is that the tactical and operational planning level problems can not be solved separately,

but have to be considered simultaneously.

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The problem to determine the scheduled arrival and departure times under uncertainty in port

times and a predetermined sailing speed policy is considered in Qi and Song (2012) and Wang

and Meng (2012). Qi and Song (2012) provide some useful insights in the optimal schedule under

100% service level constraints. Wang and Meng (2012) formulate the problem as a two-stage

stochastic programming problem and solve it using sample average approximation. They are

able to �nd solutions with an objective value within 1.5% of the optimal solution in less than

one hour.

3 Terminal operations

Liner shipping operations are closely related to terminal operations and decisions about ships

cannot be taken while disregarding their e�ects on terminals. In fact, terminals are the largest

bottleneck for shipping. It is important to have the right berth slots and to be loaded and

unloaded quickly and in a predictable manner. There are many ports all over the world with a

large number of ships waiting in front of the harbour to be allowed to berth. Accordingly we will

discuss those terminals operations aspects which directly a�ect shipping, viz. berth scheduling,

crane allocation and container stacking.

3.1 Berth scheduling

Both at a tactical as well as at an operational level, liner shipping schedules are made while

taking berth availability into account. On a tactical level, when designing the liner shipping

routes, agreements are made with terminals on berth availability and productivity (how many

cranes and crane teams will be employed and how many container moves will be done per hour).

This enables the shipping line to calculate the port time of his ships and to complete the ship

scheduling. Naturally bu�er times are incorporated in the schedule and in the berth schedule

in order to take care of schedule deviations. Quite often agreements are made on demurrage

charges (penalties related to delayed cargoes) if terminals need more time or if the shipping line

arrives too late at the terminal. At the operational level the berth schedule is adjusted according

to actual information. Quite often liner ships are too late. In 2015 Drewry shipping reports that

ships are on average one day late. So the berth schedule is updated at a relatively short term

(2 weeks) to take care of changing circumstances, while taking the tactical berth planning as a

start.

3.2 Crane allocation

One level deeper than berth scheduling is the crane allocation. Cranes are used to move a

container from the quay to the location where it will stored on the ship and vice versa. The

storage locations on the ships are called locks. As container ships typically visit many ports,

the cargo destined for a particular port will be distributed over many holds in the ship. After

unloading, a ship will load cargo for several destinations which all have to be put in di�erent

ship holds. As a result the scheduling of the cranes is a di�cult stochastic problem (handling

times of containers are quite variable). The last crane to �nish determines the moment when the

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ship can leave and hence the port time. A good balancing of the workload between the cranes

is therefore necessary, but also very di�cult to achieve. Another complication comes from the

fact that port workers often work in shifts with �xed starting and end times, and a terminal will

have to accommodate these restrictions.

3.3 Container stacking

A �nal aspect we like to mention is the stacking of the containers in the yard. The issue is not

only that containers are stacked on top of each other, which complicates the retrieval of a bottom

container, it is also the location on the yard of the containers to be loaded. If a ship moors right

before the place where its (to be loaded) containers are located, then travel distances to the

quay cranes are short and no bottlenecks are likely to occur. However, if a ship (due to delay or

congestion at the berths) berths somewhere else, or if containers are spread out over the yard,

then the terminal has to transport the containers over longer distances by which the loading

could potentially be delayed. Container stacking is closely related to stowage planning, as the

latter determines the order in which containers are to be loaded. In a perfect world one can

take the order in which containers are stacked into account while making the load planning, but

that creates a very complex problem, which also su�er from the variations in the loading. Hence

costly reshu�es, where top containers are placed somewhere else to retrieve bottom containers

are needed in large quantities.

4 Geographical bottlenecks

Canal restrictions form the main geographical bottlenecks for container ships. The Suez Canal

and the Panama Canal are two well known canals imposing restrictions on container ships. The

type of restrictions may di�er between di�erent canals. The main restriction imposed by the

Suez Canal is for example the compulsory convoy passage through the canal. This results in long

waiting times if a container ship misses the planned convoy. The Panama Canal on the other

hand, imposes limits on the size of ships that want to sail through the canal.

Two other examples of geographical bottlenecks are the Strait of Malacca and the Gulf of Aden.

These waterways are narrow, but are strategically important locations for the world trade, making

them vulnerable to piracy.

Finally, ports may also impose geographical bottlenecks. Large ships might not be able to access

certain ports, because the access ways have tight draft restrictions.

5 Operations research in liner shipping

In 1983, Ronen provided the �rst overview paper on the contribution of operations research meth-

ods in ship routing and scheduling. Since this �rst paper, every decade a follow-up overview paper

appeared reviewing new research conducted in that decade (Ronen 1993, Christiansen et al. 2004,

2013). Initially, the reviews were mainly focused on the ship routing and scheduling problem, but

more and more other shipping problems are included in these reviews. Furthermore, Christiansen

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et al. (2007) provides an extensive overview of maritime transportation problems. Finally, Meng

et al. (2014) give an overview of research related to container routing and scheduling in the liner

shipping industry in the last thirty years. The number of citations in these reviews has increased

fast, showing the increasing interest in operations research in liner shipping problems.

Liners usually face complex problems, because the above discussed decision problems cannot be

seen separately from each other and because problem instances are usually large. For example,

when a liner company wants to determine its service network, it has to consider which e�ect

the included routes will have on the cargo routing problem. The solution to the cargo routing

problem depends on the underlying network and will in�uence the pro�t of that network. This

increases the complexity of the problems faced by liners, since they need to solve multiple decision

problems simultaneously. Furthermore, the number of ports that need to be included in a network

is usually large (it can easily contain over 100 ports). The Indonesian case study in the next

section will show that designing a network for only six ports is already quite di�cult. Liner

companies used to solve these problems manually, but in the last years computerized decision

support systems became available. A well-known example of a successful decision support system

is TurboRouter, a tool for optimizing vessel �eet scheduling (Fagerholt and Lindstad 2007).

6 Case study: Indonesia

Shipping is an important mode of transport in Indonesia because the country consists of many

islands. Figure 1 shows six main ports in Indonesia. The six ports are located on �ve di�erent

islands of Indonesia, hence transport over land is only possible between Jakarta and Surabaya.

Transportation between all other combinations of these cities is only possible by sea or air.

Belawan

JakartaSurabaya

Banjarmasin

Makassar

Sorong

Figure 1: Location of six main ports in Indonesia

We will use the Indonesian case to illustrate some of the decision problems introduced in Section 2.

Thereto, we will assume that Table 1 gives the expected weekly demand in TEUs between the

Indonesian ports. The last column and row give the row and column sums, denoting respectively

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the total supply from and demand to a port. The supply and demand values denote the number

of containers leaving and arriving in the port respectively. The di�erence between demand and

supply indicates how many empty containers have to be repositioned from or to the port. For

the six ports in Indonesia, the empty container repositioning problem is of limited importance,

since there are no large di�erences between supply and demand.

Belawan Jakarta Surabaya Banjarmasin Makassar Sorong Supply

Belawan - 6,500 1,000 100 75 25 7,700

Jakarta 6,750 - 2,000 4,000 2,800 450 16,000

Surabaya 1,000 2,500 - 3,750 4,800 2,150 14,200

Banjarmasin 100 3,600 3,500 - 10 0 7,210

Makassar 100 3,500 4,000 75 - 0 7,675

Sorong 50 650 2,000 0 0 - 2,700

Demand 8,000 16,750 12,500 7,925 7,685 2,625 55,485

Table 1: Expected weekly demand in TEU between the Indonesian ports (Source: own calcula-tions)

Table 1 shows that Jakarta and Surabaya are the two ports with the largest container throughput,

while trade with Sorong is relatively small. In this speci�c case, this might lead to problems, since

Sorong is also located relatively far away from the other ports. Liner shipping companies prefer

to o�er services calling at the ports of Jakarta and Surabaya and consider it too costly to call at

Sorong. By charging higher prices for containers that have to be transported from or to the port

of Sorong, liners can make stops at Sorong more attractive. Hence, the liner company may use

the pricing strategy to ensure that services calling at Sorong will also be bene�cial. However, to

determine exactly which prices they have to charge in order to maximize their pro�t, the liner

company needs more details on the cost structure of the network they will provide.

Figure 2 shows examples of a hub and feeder system, a circular route, a butter�y route and a

pendulum route calling at the six Indonesian ports. In the hub and feeder system of Figure 2a

the port of Surabaya is the hub port, while Belawan, Jakarta, Banjarmasin, Makassar and Sorong

are feeder ports. The route Surabaya - Jakarta - Belawan - Surabaya is referred to as F1. F2 is a

direct feeder route between Surabaya and Sorong. The third feeder route, F3, calls at Surabaya,

Banjarmasin and Makassar after which it returns to Surabaya. The circular route in Figure 2b

has as characteristic that each port is called at exactly once during the round tour. Figure 2c

shows the butter�y route Belawan - Surabaya - Banjarmasin - Makassar - Sorong - Surabaya -

Jakarta - Belawan on which Surabaya is visited twice. Finally, in the pendulum route of Figure

2d all ports are visited twice, only the second time in reversed order.

12

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Soron

g

F1

F3

F2

(a)Example

ofahubandfeeder

system

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Sorong

(b)Example

ofacircularroute

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Soron

g

(c)Example

ofabutter�yroute

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Sorong

(d)Example

ofapendulum

route

Figure

2:Route

examplesfortheIndonesianports

13

Belawan Jakarta Surabaya Banjarmasin Makassar Sorong

Belawan - 1,064 1,488 1,430 1,708 2,807Jakarta 1,064 - 438 614 806 2,102Surabaya 1,488 438 - 328 520 1,816Banjarmasin 1,430 614 328 - 353 1,577Makassar 1,708 806 520 353 - 1,375Sorong 2,807 2,102 1,816 1,577 1,375 -

Table 2: Distances between the Indonesian ports in nmi (Source: www.ports.com/sea-route/)

Table 2 shows the distances in nautical miles (nmi) between the six Indonesian ports and Table 3

provides some characteristics of �ve ship types. Types 1, 2, 3 and 5 are obtained from Brouer

et al. (2014a), while Type 4 is suggested by the Indonesian government and costs are obtained

using interpolation. Note that the fuel usage in ton/day of Type 4 is larger than the usage

of Type 5, because Type 4 has a higher design speed than Type 5. These data can be used

to get some insight in the route cost using di�erent ship types and network structures. In the

calculations we use a simpli�ed version of the fuel cost function as provided in Brouer et al.

(2014a):

Fs(v) = 600 ·(v

v∗s

)3

· fs (1)

Here, Fs(v) denotes the fuel cost in USD per day for a ship of type s sailing at a speed of v knots

(nmi/hour). v∗s and fs are the design speed and fuel consumption in ton per day of a ship of

type s sailing at design speed and can be found in Table 3. Remark that the bunker cost varies

over time, but is assumed to be constant and equal to 600 USD per ton in this study (Brouer

et al. 2014a). Table 4 shows the route distance in nautical miles, the duration in weeks, the

frequency, the number of ships required to obtain the frequency and the sailing speed in knots

for each route. Distances can be found by adding the distances of the individual sea legs, while

the duration and frequency are manually �xed in this example.

Ship Capacity Charter cost Draft Min speed Design speed Max speed Fuel usagetype (TEU) (USD/day) (m) (knots) (knots) (knots) (ton/day)

1 900 5,000 8 10 12 14 18.82 1,600 8,000 9.5 10 14 17 23.73 2,400 11,000 12 12 18 19 52.54 3,500 16,000 12 12 18 20 60.05 4,800 21,000 11 12 16 22 57.4

Table 3: Data of the ship characteristics (Source: Brouer et al. 2014a)

In liner shipping it is common to use weekly port calls at a route. Route durations are typically

an integer number of weeks such that an integer number of ships is needed to sail this route.

For example, a route with duration three weeks and which is sailed by three ships, ensures a

weekly frequency. Given the duration and frequency, the number of required ships can be found

by taking the product of these two values.

14

Route Distance Duration Frequency Required Speed(nmi) (weeks) (per week) ships (knots)

F1 2,990 2 1 2 11.33*

F2 3,632 2 1 2 12.61F3 1,201 1 1 1 12.51Circular 6,476 4 1 4 12.27Butter�y 6,862 4 1 4 13.62Pendulum 7,802 5 1 5 13.55

Table 4: Route characteristics for the di�erent ships (an * indicated that the speed is outsidethe feasible range for some ship types)

10 12 14 16 18 20 220

50

100

150

Speed (knots)

Fuel

cost

(USD/n

m)

Ship 1Ship 2Ship 3Ship 4Ship 5

Figure 3: Fuel cost in USD per nautical mile

Figure 3 shows the fuel price in USD per nautical mile at di�erent speeds for the �ve ship

types. The fuel price is a convex function, meaning that when the speed is doubled, the fuel

cost per nautical mile is more than doubled. Hence, a constant sailing speed during the route

will minimize the fuel cost. The speed is calculated under the assumption that every port call

takes 24 hours and the durations as given in Table 4. The following formula can then be used to

determine the speed on each route:

v =δ

168 · t− 24 · n, (2)

where δ is the route distance in nautical miles, t the route duration in weeks and n the number

of port calls on the route. An ∗ in the column denoting the speed of Table 4 indicates that the

speed is outside the feasible speed range for some ship types. The frequency is chosen in such a

way that it is feasible for each ship type when sailing at maximum speed. Hence, the necessary

speed can only be lower than the minimum speed of the ship type, in which case the ship will

sail at minimum speed and will wait in one of the ports to obtain a weekly frequency.

Table 5 shows the weekly cost in USD for each of the routes given the frequency and duration

as given in Table 4. The route costs consist of three components: the �xed ship costs, the port

15

call costs and the fuel costs. When a liner company needs three ships to satisfy the required

route duration and frequency, it bears weekly the �xed ship costs of all these three ships. Hence,

the �xed ship cost is given by 7 · S · cfs , with S the number of required ships and cfs the daily

�xed ship cost of type s, which can be found in Table 3. The port call cost is the sum of the

port fees of the ports visited on the route. If we assume that all port fees are the same, the port

call cost is given by Fp · n · q, where Fp is the port fee per port visit and q the route frequency.In this example, we assume that Fp = 650 USD. The fuel cost is given by the product of the

frequency, the number of days that a ship needs to sail one round tour and the fuel cost per

day: q · δ24·v ·Fs(v), where Fs(v) is the fuel cost in USD per day when sailing at speed v as given

by (1). Consider a liner route with a duration of two weeks to which four ships are allocated.

Each port on the route will then be called twice a week, resulting in a frequency of twice a week.

Each ship needs two full weeks to sail a round tour, so in one week it will sail half of the route.

Since there are four ships allocated to the route, in total two full round tours are made during

a week (since the frequency is two). This explains the multiplication with the frequency in the

fuel and port call cost. The total route cost in USD per week is now given by:

crs = 7 · S · cfs + q · δ

24 · v· Fs(v) + Fp · n · q, (3)

where crs is the route cost in USD per week for a ship of type s. Doubling the capacity of a ship

will not result in a doubling of the weekly route cost. This illustrates the e�ect of economies of

scale: larger ships will in general have higher total costs, but lower costs per TEU, which is also

exempli�ed in Table 6 by showing the weekly route cost per TEU under the assumption that the

ship is fully utilized. The table also shows that the e�ect of economies of scale can di�er quite a

lot between ship types.

Cost in USD/week

Route/Ship Type 1 Type 2 Type 3 Type 4 Type 5

F1 176,268 196,765 252,848 336,691 446,793F2 228,411 238,026 285,297 373,868 497,669F3 88,076 98,537 121,253 162,296 214,803Circular 408,876 438,257 531,147 702,468 933,207Butter�y 490,525 503,210 598,818 779,713 1,038,189Pendulum 571,488 596,234 714,297 935,318 1,243,643

Table 5: Route cost per week for the duration and frequency as given in Table 4

16

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Soron

g

16,950

17,250

18,000

15,600

14,885

14,875

2,70

0

2,62

5

Cap

acityroutes:

F1:

18,200

F2:

3,50

0

F3:

16,000

(a)Utilizedcapacities

inTEU

forthehubandfeeder

system

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Sorong

27,460

26,710

28,41028,485

28,475

27,760

Cap

acity:28

,800

(b)Utilizedcapacities

inTEU

forthecircularroute

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Sorong

17,025

17,325

18,075

18,225

17,510

17,500

17,575

Cap

acity:19

,200

(c)Utilizedcapacities

inTEU

forthebutter�yroute

Belaw

an

Jakarta

Surabaya

Ban

jarm

asin

Makassar

Sorong

7,70

0

8,00

0

10,450

11,500

11,625

9,86

0

10,310

10,375

2,625

2,70

0

Cap

acity:11

,800

(d)Utilizedcapacities

inTEU

forthependulum

route

Figure

4:Route

exampleswithutilizedcapacities

17

Cost per TEU in USD/week

Route/Ship Type 1 Type 2 Type 3 Type 4 Type 5

F1 196 123 105 96 93F2 254 149 119 107 104F3 98 62 51 46 45Circular 454 274 221 201 194Butter�y 545 315 250 223 216Pendulum 635 373 298 267 259

Table 6: Economies of scale in ship size at full utilization

The disadvantage of the circular route is that the capacity can not be utilized e�ciently. When

containers from for example Surabaya to Jakarta are transported using the circular route, they

will be on board of the ship on all sea legs except the leg from Jakarta to Surabaya. Butter�y

routes are better able to utilize the available capacity, since some ports are visited twice on a

round tour. In the butter�y route, the ports of Surabaya and Jakarta are visited directly after

each other, such that the containers are only on board during one sea leg of the route. The

pendulum route visits all ports twice, hence it needs the lowest capacity. In a hub and feeder

network, usually many ports are connected by only one or a few sea legs. This ensures that hub

and feeder networks are able to utilize the available capacity very e�ciently. Figure 4 shows

the utilized capacity at each sea leg in the four di�erent route networks under the assumption

that all demand has to be satis�ed using only the given network. For the butter�y route, we

assumed that the containers that have to be transported from Makassar to Banjarmasin will

stay on board of the ship during the route segment Surabaya - Jakarta - Belawan - Jakarta

- Surabaya. Alternatively, these containers can be unloaded during the �rst call at Surabaya

and loaded again during the second port call at Surabaya in which case transshipment costs at

Surabaya are incurred. The utilized capacities are found by adding all container �ows that need

to traverse the given sea leg in order to arrive at their destination. Table 7 shows the required

capacity in TEU for each route, the number of port calls per week for each ship type in order

to have enough capacity to satisfy all demand, the available capacity in TEU using these ship

types and the total route costs in USD per week. The required capacity is found by taking the

maximum utilized capacity of the route. Next, we make a combination of ship types such that

enough capacity is available at each route. Given these ship allocations, the total route cost can

be found by multiplying the weekly route cost for a ship type by the number of port calls per

week divided by the route frequency. Note that the type and number of ships needed vary a lot

between the three di�erent route structures. For the hub and feeder system, 2 ·1+1 ·1 = 3 ships

of type 2 (since feeder route 1 has a duration of 2 weeks and feeder route 3 has a duration of

one week), 2 · 2 + 2 · 1 = 6 ships of type 4 and 2 · 2 + 1 · 3 = 7 ships of type 5 are needed. The

circular route uses 4 · 6 = 24 ships of type 5, while the butter�y route uses 4 · 4 = 16 ships of

type 5. Finally, the pendulum route uses 5 · 2 = 10 ships of type 4 and 5 · 1 = 5 ships of type 5.

Hence, the optimal solution to the �eet size and mix problem is highly dependent on the network

structure.

Table 7 also shows the total network cost for the hub and feeder system, the circular route, the

18

Route Req. cap. Port calls per week Av. cap. Cost

(TEU) Type 1 Type 2 Type 3 Type 4 Type 5 (TEU) (USD/week)

F1 18,000 0 1 0 2 2 18,200 1,763,733F2 2,700 0 0 0 1 0 3,500 373,868F3 15,600 0 1 0 0 3 16,000 742,948HF-Total 2,880,549

Circular 28,485 0 0 0 0 6 28,800 5,599,239

Butter�y 18,225 0 0 0 0 4 19,200 4,152,755

Pendulum 11,625 0 0 0 2 1 11,800 3,114,280

Table 7: Network cost per week when shipping all demand

butter�y route and the pendulum route. The table indicates that the hub and feeder system

and the pendulum route are by far the cheapest choices of networks in this example. They

both cost approximately 3 million USD per week, while the circular and butter�y routes cost

respectively about 5.5 and 4 million USD per week. One remark has to be made: in the hub and

feeder system, a lot of containers need to be transshipped, adding additional costs that are not

included in this example. In total 15,450 containers have to be transshipped per week in the hub

and feeder system. If a transshipment costs for example 100 USD per container, the total cost

of the hub and feeder system will rise to almost 4.5 million USD per week. Hence, the hub and

feeder system will then have higher costs than the butter�y and pendulum routes. Of course,

one could also make route networks with combinations of these routes, which might be more cost

e�cient.

The good performance of the hub and feeder system and pendulum route is (partly) caused

because of the better utilization of capacity in the hub and feeder system. Another advantage

of hub and feeder systems is that liners can allocate di�erent ship types to the di�erent types of

routes. Feeder ports usually have less demand than hub ports, hence it makes sense to allocate

smaller ships to the feeder routes than to the main routes. If all ports are visited on similar

routes, like circular, butter�y and pendulum routes, all these ports are visited by the same ship

type. Hence, large ships might visit very low demand ports if these ships are able to berth in the

smaller ports (smaller ports might have stricter draft restrictions than hub ports). Otherwise

many small ships are needed in order to satisfy the demand of the large ports. However, a

disadvantage of hub and feeder networks is that usually many transshipments are needed in

order to satisfy the demand, which increases both the transportation price and transit time. In

airline passenger transport, hub and feeder systems are very popular; an important reason for

this is that transshipments are made by passengers at no apparent cost.

Finally, we determine the pro�t and e�ciency of the networks when we assume that each con-

tainer will generate a revenue of 200 USD if it is transported, (un)loading and transshipment

costs are all 40 USD per container. Recently, the problem is also studied on request of the In-

donesian government, resulting in a single pendulum route to be sailed. This pendulum route is

also known under the name Pendulum Nusantara. We use the mixed integer programming model

proposed in Mulder and Dekker (2016) to determine the optimal route network given an initial

set of routes. Routes are constructed in the following way using the ordering of ports used for

19

Belawan

JakartaSurabaya

Banjarmasin

Makassar

Sorong

Capacity: 3,500

(a) Route 1

Belawan

JakartaSurabaya

Banjarmasin

Makassar

Sorong

Capacity: 3,50

(b) Route 2

Belawan

JakartaSurabaya

Banjarmasin

Makassar

Sorong

Capacity: 4,800

(c) Route 3

Figure 5: Optimal route network

20

the pendulum route. Ports may be visited at most twice during a route: once on the eastbound

trip and once on the westbound trip. All feasible routes are generated and given as input to the

mixed integer programming problem, which makes use of a path formulation to solve the cargo

allocation. Table 8 shows the pro�ts of these networks and Figure 5 shows optimal route net-

work. We see that the pendulum route performs indeed better than the hub-and-feeder network

and the butter�y and circular routes. The e�ciency of the networks is measured by the shipped

distance in nmi per TEU. The shipped distance for direct shipping is equal to 836.57 nmi/TEU.

Table 8 shows that the pendulum and optimal networks are both e�cient networks with respect

to shipped distance. Furthermore, the hub-and-feeder network is much more e�cient than the

circular and butter�y routes as expected.

The optimal route network as shown in Figure 5 consists of two pendulum routes (Routes 2 and

3) and a non-stop service (Route 1), which is a special type of pendulum route with only two

port calls. The pendulum route structure ensures e�cient transportation between all demand

pairs. All routes have a frequency of once a week, reducing the number of required ships.

Network Shipped distance (nmi/TEU) Pro�t (USD)

Hub-and-feeder 1,428.06 3,159,651Circular 3,269.79 1,058,961Butter�y 2,227.57 2,284,445Pendulum 996.80 3,642,916Optimal 852.66 4,897,109

Table 8: E�ciency and pro�t of the di�erent networks

7 Conclusion

Maritime transportation is very important in the world economy. The types of operations in

the shipping market are distinguished in tramp, industrial and liner shipping. This chapter

considers decision problems that occur in the operations of container liner shipping companies.

The decision problems can be distinguished in three di�erent planning levels: strategic, tactical

and operational. The strategic planning level consists of long term decision problems, while the

tactical and operational planning levels contain respectively medium and short term problems.

This chapter discusses the �eet size and mix and market trade selection problem on the strategic

level, network design, pricing and empty container repositioning on the tactical level and cargo

routing, disruption management, revenue management and stowage planning on the operational

planning level. Furthermore, sailing speed and bunkering optimization and robust schedule

design are covered. These decision problems are related to both the tactical and the operational

problem. The chapter also introduces three decision problems in the terminal operations that

are related to container liner shipping: berth scheduling, crane allocation and container stacking.

A case study is used to explain the concepts of the problems in more detail. The case study is

based on six main ports in Indonesia. Three networks with di�erent structures (hub and feeder

system, pendulum route and butter�y route) are proposed. Calculations show that the cost

21

of the hub and feeder system is much smaller than the cost of the other two networks when

transshipment costs are not considered. Explanations for this result are that capacity is better

utilized in hub and feeder systems compared to the other network structures and ships can be

chosen more freely, since more shorter routes are used. When transshipments are charged at 100

USD per container, the hub and feeder system performs comparable to the butter�y route and

still considerably better than the pendulum route.

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24